Clare, who I work with, is organising the London marathon this year

I have read a rule of relative clause reductions, and it states,

2. We can use participles when reducing the sentence.

a) Present Participle "V-ing" (simultaneous)

1) We stood on the bridge which connects the two halves of the city.

//the relative clause is a defining clause.

a) We stood on the bridge connecting the two halves of the city.

//omitting by replacing them with present participle

Since the example has stated only about a sentence that contains a defining clause, that makes me curious whether we can also omit the pronouns with or without the to be verb in the non-defining relative clauses of the sentences below,

2) Clare, who works with me, is organising the London marathon this year.

3) Clare, who is working with me, is organising the London marathon this year.

into this sentence:

b) Clare, working with me, is organising the London marathon this year.

If so, why?

Solution 1:

As Kris points out in a comment, this doesn't work.

Clare, who works with me, is organising the London marathon this year.

Clare is organising. I don't necessarily have any connection with the event. It happens that Clare works with me.

Clare, who is working with me, is organising the London marathon this year.

This sentence is slightly ambiguous but, as a native speaker, I would interpret it to mean, "Clare, who currently is working with me, is organising the London marathon this year." Again I may not have any involvement with the marathon.

Clare, working with me, is organising the London marathon this year.

This indicates unambiguously that Clare and I are working together on the organisation of the marathon.

Explanation

In the first two, "who is working with me"is adjectival and describes something about Clare.

In the last sentence, "working with me" is adverbial to the verb 'organising'.